You have found the following ages (in years) of all 4 sloths at your local zoo: $ 13,\enspace 10,\enspace 4,\enspace 10$ What is the average age of the sloths at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we have data for all 4 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $4$ ages and divide by $4$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\mu} = \dfrac{13 + 10 + 4 + 10}{{4}} = {9.3\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $13$ years $3.7$ years $13.69$ years $^2$ $10$ years $0.7$ years $0.49$ years $^2$ $4$ years $-5.3$ years $28.09$ years $^2$ $10$ years $0.7$ years $0.49$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{13.69} + {0.49} + {28.09} + {0.49}} {{4}} $ $ {\sigma^2} = \dfrac{{42.76}}{{4}} = {10.69\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{10.69\text{ years}^2}} = {3.3\text{ years}} $ The average sloth at the zoo is 9.3 years old. There is a standard deviation of 3.3 years.